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Roughly, the Socle Lemma refers to the idea that, in a definable group of
finite rank, a type with trivial stabiliser should be internal to rank 1
types. Hrushovski's original argument applied only to rigid abelian groups of
finite rank. Under the assumption of the Canonical Base Property (CBP), Pillay
and Ziegler \cite{PZ} gave a version for arbitrary finite rank groups.

In the paper under review, the authors prove a version of the Socle Lemma for
rigid groups in simple theories, with no finite rank hypothesis. Roughly, a
group is rigid if it has no definable families of subgroups. The Socle
statement becomes that if $G$ is rigid and analysable in a family $\Sigma$ of
partial types, then any type of an element of $G$ with bounded stabiliser is
almost internal to $\Sigma$.

This is to be contrasted with the result of Hrushovski-Palacin-Pillay
\cite{HPP} that a stable non-multidimensional theory in which all Galois
groups are rigid has the CBP, and hence that a Socle Lemma holds.

There is no obvious relation between the assumption that the Galois groups are
rigid and the assumption that $G$ in the statement of the Socle Lemma is
rigid, nor is there an indication that rigidity of $G$ implies any version of
CBP. Nonetheless, the authors of the present paper take the opportunity to
use a result of Chatzidakis to mildly strengthen the HPP result, removing the
assumption of non-multidimensionality.

The proof of the main Socle result involves an analysis of the properties of
stabilisers and rigidity in groups (hyper)definable in simple theories. A
knowledge of the basics of simplicity theory is assumed.

\begin{thebibliography}{HPP13}

\bibitem[HPP13]{HPP}
Ehud Hrushovski, Daniel Palac{\'{\i}}n, and Anand Pillay.
\newblock On the canonical base property.
\newblock {\em Selecta Math. (N.S.)}, 19(4):865--877, 2013.

\bibitem[PZ03]{PZ}
Anand Pillay and Martin Ziegler.
\newblock Jet spaces of varieties over differential and difference fields.
\newblock {\em Selecta Math. (N.S.)}, 9(4):579--599, 2003.

\end{thebibliography}
%\bibliography{rigidity}

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